{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ZXEYDX6VD4U4NMTXVDVMXHGWB3","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"42ccd6acb995d951746692f115219bd78140bed832b18a857926aa770c5f4796","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-25T16:59:45Z","title_canon_sha256":"ba670833cff5659c28c9e2e66c335b73cf2791a993bd0f543841b001a43aed8c"},"schema_version":"1.0","source":{"id":"1201.5324","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.5324","created_at":"2026-05-18T04:03:55Z"},{"alias_kind":"arxiv_version","alias_value":"1201.5324v1","created_at":"2026-05-18T04:03:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.5324","created_at":"2026-05-18T04:03:55Z"},{"alias_kind":"pith_short_12","alias_value":"ZXEYDX6VD4U4","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZXEYDX6VD4U4NMTX","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZXEYDX6V","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:56aa70d0fc48c6d50854dea7298bd12ef1fdee679b0cd4c40a5aeb4625948250","target":"graph","created_at":"2026-05-18T04:03:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"This paper deals with higher gradient integrability for $\\sigma$-harmonic functions $u$ with discontinuous coefficients $\\sigma$, i.e. weak solutions of $\\div(\\sigma \\nabla u) = 0$.\n  We focus on two-phase conductivities, and study the higher integrability of the corresponding gradient field $|\\nabla u|$. The gradient field and its integrability clearly depend on the geometry, i.e., on the phases arrangement. We find the optimal integrability exponent of the gradient field corresponding to any pair $\\{\\sigma_1,\\sigma_2\\}$ of positive definite matrices, i.e., the worst among all possible microg","authors_text":"Marcello Ponsiglione, Mariapia Palombaro, Vincenzo Nesi","cross_cats":["math.CV"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-25T16:59:45Z","title":"Gradient integrability and rigidity results for two-phase conductivities in dimension two"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.5324","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:96a66062b0412f4783697c1b61a87a543ad6a7d5e0442788c25ba10517856c3a","target":"record","created_at":"2026-05-18T04:03:55Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"42ccd6acb995d951746692f115219bd78140bed832b18a857926aa770c5f4796","cross_cats_sorted":["math.CV"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-25T16:59:45Z","title_canon_sha256":"ba670833cff5659c28c9e2e66c335b73cf2791a993bd0f543841b001a43aed8c"},"schema_version":"1.0","source":{"id":"1201.5324","kind":"arxiv","version":1}},"canonical_sha256":"cdc981dfd51f29c6b277a8eacb9cd60edc1dc6be039af07ff1538fdd5b41aca3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cdc981dfd51f29c6b277a8eacb9cd60edc1dc6be039af07ff1538fdd5b41aca3","first_computed_at":"2026-05-18T04:03:55.044839Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:03:55.044839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ZZZk2R+BwCZbk7msZqtZYuWk2W+yDMohlkORm6yfCzI0w/28n54bwWPztbtpzDPpDCCePCNyiHHIOCYd5fCpCg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:03:55.045522Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.5324","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:96a66062b0412f4783697c1b61a87a543ad6a7d5e0442788c25ba10517856c3a","sha256:56aa70d0fc48c6d50854dea7298bd12ef1fdee679b0cd4c40a5aeb4625948250"],"state_sha256":"2299b864400b7b74a79d799d1693d998dc80b2bdef9bef7079da50abc5cffad7"}